Global Financial Management

Financial Mathematics

Copyright 1997 by Campbell R. Harvey and Stephen Gray. All rights reserved. No part of this lecture may be reproduced without the permission of the authors.

Latest Revision: December 1996

Lecture 0. The Time Value of Money

Would you rather receive $1000 today -- or next year? Obviously, you would want the money today. You could bank the money today and it would be worth more than $1000 next year. The most basic principle of Finance is that a dollar today is worth more than a dollar in the future.

We will explore the answer to two questions: "What is the value next year of $1000 received today?" and "What is the value today of $1000 received next year?" The first question asks us to find the future value of money received now. The second question asks us to find the present value of money received in the future.

Before answering these questions, we need to establish some notation:

V0 Present Value
Vn Future Value at the end of n periods
n The number of periods.
i Effective periodic interest rate, or effective interest rate per compounding period
R Nominal interest rate, also called the "annual percentage rate" (APR) or the "stated interest rate"
m The number of compounding periods per year
t The number of years. Note t = n/m
r The effective annual interest rate, also called the true annual interest rate

The effective periodic interest rate, i, equals the nominal interest rate R divided by the number of compounding periods per year.

The effective (or true) annual interest rate, r, is the annually compounded interest rate which is equal to the effective periodic rate, i, compounded for m periods.

r = (1+i)m-1 = (1+R/m)m-1

When interest is compounded continuously.

r = eR

where e is the natural exponent, which is approximately 2.718.

Click here for a proof.

0.1 Future Value -- Single Cash Flow
     Annual Compounding

The future value formula is derived by example. Suppose that you have a deposit of $1000 which is paid 4% annually. What is the deposit

Year Beginning Balance Interest Payment Ending Balance Formula
1 1000.00 40.00 1040.00 1000 (1.04)
2 1040.00 41.60 1081.60 1000 (1.04)2
3 1081.60 43.26 1124.86 1000 (1.04)3
4 1124.86 44.99 1169.85 1000 (1.04)4
5 1169.85 46.79 1216.64 1000 (1.04)5
6 1216.64 48.67 1265.31 1000 (1.04)6

The formula for future value when interest is compounded annually is straight-forward from this example:

Vn = V0 (1+R)n

0.2  Continuous and Discrete Time Compounding

Suppose that another bank offers you the same nominal interest rate of 4%, but offers to compound the interest every six months. Would you prefer this deal?

The deposit will pay 2% every six months. The following table shows how this investment will grow:

Year Beginning Balance Interest Payment Ending Balance Formula
0.5 1000.00 20.00 1020.00 1000 (1.02)
1.0 1020.00 20.40 1040.40 1000 (1.02)2
1.5 1040.40 20.81 1061.21 1000 (1.02)3
2.0 1061.21 21.22 1082.43 1000 (1.02)4
2.5 1082.43 21.65 1104.08 1000 (1.02)5
3.0 1104.08 22.08 1126.16 1000 (1.02)6
3.5 1126.16 22.52 1148.68 1000 (1.02)7
4.0 1148.68 22.97 1171.65 1000 (1.02)8
4.5 1171.65 23.43 1195.08 1000 (1.02)9
5.0 1195.08 23.90 1218.98 1000 (1.02)10
5.5 1218.98 24.38 1243.36 1000 (1.02)11
6.0 1243.36 24.87 1268.23 1000 (1.02)12

This investment will be worth an extra $0.40 in the first year, and an extra $2.92 at the end of the sixth year, because of compounding twice per year. From this table, we can also postulate a generalized formula for future value, when there are m compounding periods per year.

Vn = V0 (1+R/m)n = V0 (1+i)n

If the interest is compounded continuously, then the formula becomes:

Vn = V0 eRt

While the nominal rate on these deposits remains the same, the effective annual rate changes. The effective annual rate can be used to compare investments with different compounding periods.

Example 0.3

Suppose a bank offers a nominal interest rate of 4% (R = 0.04) on your savings deposit. The following table illustrates the different effective or true interest rates depending on how many times the interest is compounded each year.

Compounding Formula Effective Annual Rate
Annually r = (1+0.04/1)1-1 4.00000%
Semiannually r = (1+0.04/2)2-1 4.04000%
Quarterly r = (1+0.04/4)4-1 4.06040%
Monthly r = (1+0.04/12)12-1 4.07425%
Weekly r = (1+0.04/52)52-1 4.07947%
Daily r = (1+0.04/365)365-1 4.08084%
Hourly r = (1+0.04/8760)8760-1 4.08088%
Continuous r = e0.04 4.08108%

So, the investor will always prefer more compounding periods to less. The continuous time rate of interest is always higher than the periodic interest rate.

Click here for an optional discussion of compounding periods

0.4 Money Multiplier

The term

Mn = (1+R/m)n

for discrete compounding, or

Mt = eRt

for continuous compounding, is sometimes referred to as the money multiplier. As the name implies, the money multiplier measures the factor by which your money multiplies in the future, given a nominal rate R and a maturity of n periods. Often, the return on investment depends upon the length of time the money is tied up. Consider a schedule of bank interest rates. The 1-5 year rates were quoted from Wachovia.

Investment Period Money Rate (R) Multiplier (Mt)
1 Year 8.150 1.0815000
2 Years 8.200 1.1707240
3 Years 8.350 1.2719989
4 Years 8.400 1.3807566
5 Years 8.500 1.5036567
10 Years 9.000 2.3673637
15 Years 9.000 3.6424824
20 Years 9.000 5.6055107
30 Years 9.000 13.267678

Note that the money market multiplier increases exponentially with longer time to maturity. Furthermore, the rate of growth depends upon the interest rate.

0.5  Present Value -- Single Cash Flow

The Present Value formula can be derived from the formula for the future value. Suppose that we know the future value (Vn) of an investment. The present value of that investment (V0) is easily calculated. From the formula for future value, we know:

Vn = V0 (1+R/m)n

Divide both sides by the money multiplier to get the present value:

V0 = Vn (1+R/m)-n

For continuous compounding, we have the following formula:

V0 = Vn e-Rt

Example 0.6

If you wish to provide $200,000 for your newborn's college education, how much should you invest now if the interest rate is 8% compounded annually? That is, what is the present value of $200,000 in 18 years time at 8%?

V0 = Vn (1+R/m)-n = $200,000 (1.08)-18 = $50,049.81

The formula for finding the present value of a single cash flow can be used to find the present value of a set of cash flows by finding the value of each separate flow, and adding them all together.

0.7  Annuity Valuation

An annuity is a stream of cash flows that are equally spaced in time and of equal amount. An example is a $250,000 mortgage at 9% per year, or 9%/12 = .75% per month, that is paid off with a 180 month annuity of $2,535.67. We will show how to calculate the present value of an annuity and how to determine the size of an annuity that is necessary to pay off a certain present value (like the $250,000 mortgage).

We will use the following notation:

m The number of compounding periods per year
t The number of years in annuity. Note t = n/m
n The total number of periods (n = mt)
R Annual Percentage Rate (APR) or the "stated interest rate"
a Amount of the annuity payment
An Present value of n period annuity of a dollars

Let Z be the present value of $1 at the end of one period. From the formula for the present value of a single cash flow, we know that:

This value is used to find the present value of an annuity. The present value of an n period annuity of a dollars is given by the following formula:

Click here for a proof.

Some people prefer another form of the formula. Recall that i = R/m. If we substitute (1+i)-1 for Z and simplify the resulting expression, we will get:

These formulas are equivalent.

Note that if the number of payments becomes infinite, then the present value of the annuity simplifies:

An example of an annuity with an infinite number of constant payments is the British consol bond. It pays a coupon at the end of each year and never matures. These bonds are called perpetuities. It is not legal to issue bonds which are perpetuities in the United States.

Example 0.8

Now we will return to the example of the $250,000 mortgage. Suppose you borrow $250,000 and repay over 15 years. The interest rate is 9% annually and payments are made monthly. The effective periodic rate of interest is 9%/12=.75% per month. Let us solve for the monthly payment a that is needed to pay off the mortgage. From our formula for the present value of the annuity, we know:

The strategy will be to substitute in for all the variables that we know (An, n, Z) and solve for the one variable that we do not know (a). First, we know that the present value must be $250,000. Second, calculate Z, the one period discount factor:

Z = 1/(1+R/m) = 1/(1+.09/12) = 0.9925558

Since n = 180,

A180 = $250,000 = a (0.9925558(1-0.9925558180))/(1-0.9925558) = a (98.59319)

Divide both sides by 98.59319,

Monthly Payment = a = (250,000/98.59319) = $2,535.67

Example 0.9

Now let's consider another example. This will highlight the idea of an amortization schedule. Suppose that $1000 is borrowed. The loan will be repaid in 5 equal annual payments (each includes interest and principal). The interest rate is 10% per annum. First, compute the one period discount factor:

Z = 1/1.10 = 0.9091

Now plug these values into the formula

A5 = 1000 = a(0.9091(1-0.90915)/(1-0.9091) = a(3.791)

Solve for a

a = 1000 / 3.791 = $263.80

Now we can check the mechanics by constructing an amortization schedule:

Year Beginning Balance 10% Interest Payment Principal Ending Balance
1 1000.00 100.00 263.80 163.80 836.20
2 836.20 83.62 263.80 180.18 656.02
3 656.02 65.60 263.80 198.20 457.82
4 457.82 45.78 263.80 218.02 239.80
5 239.80 23.98 263.80 239.82 000.00

This example illustrates the accounting implications of using an annuity. Note that there is a 2 cent rounding error.

0.10  Remaining Principal

It is often useful to know how much principal remains on a loan. Many mortgages, for example, allow the home owner to pay off the mortgage at any time. An amortization schedule, like the one above, can be used to construct the remaining principal at any time. With a thirty year mortgage, there are 360 payments. Constructing an amortization schedule can be tedious, although tools like Excel make the task easier. An easier way to find the remaining principal is to use the Annuity Formula for An -- where n is the number of periods remaining.

Example 0.11

A 30 year, $200,000.00 mortgage at a nominal rate of 8% will require monthly payments of $1,467.53. After ten years, the home owner sells the house. How much must she pay the bank in order to pay off the mortgage?

After ten years, there will still be 20 years of payments left. That means that there are 240 monthly periods. The following table shows the values needed to calculate the remaining principal:

i = 8%/12 = 0.006667 Periodic interest rate charged on mortgage.
Z = 1/1.006667% = 0.9933775 One period discount factor.
a = $1,467.53. Monthly Payment
n = 240 Number of remaining payments.

Plugging these numbers into the formula, we find that the remaining principal is equal to:

A240 = $1,467.53 (0.9933775)(1 - 0.9933775240)/ (1 - 0.9933775) =

So, she must repay $175,449.50 to the bank.

Note that this value could be either more or less than the present value of the remaining cash flows. If, for example, mortgage rates went up to 9%, then the present value of the cash flows would be less than the value of the remaining principal.

0.12  Future Value of an Annuity

It is sometimes useful to know the future value of an annuity. For example, if we are putting $100 per month into a vacation fund at 6% per year, compounded monthly, we would want to know how much money would be available after one year.

We already know how to find the present value, An of that annuity at 6% per year. We could multiply that present value by the money multiplier (1+6%/12)12 to find the future value.

Example 0.13

We can use this formula to determine the future value of the vacation fund described above. a is equal to $100, i is 6%/12 and n is 12 periods. The future value is equal to:

Summary of Formulas

r = (1+i)m-1 = (1+R/m)m-1

r = eR

Vn = V0 (1+R)n

Vn = V0 eRt

Mn = (1+R/n)n

Mn = eRt

V0 = Vn (1+R/n)-t

V0 = Vn e-Rt

An equivalent formula is: